On the non-existence of an R-labeling

TL;DR

This paper demonstrates that certain Eulerian posets cannot have R-labelings, using a structure theorem for R-labelings of the butterfly poset to establish non-existence.

Contribution

It introduces a family of Eulerian posets that lack R-labelings, revealing limitations of R-labeling applicability in poset theory.

Findings

Existence of Eulerian posets without R-labelings

A structure theorem for R-labelings of the butterfly poset

Non-existence results for R-labelings in specific posets

Abstract

We present a family of Eulerian posets which does not have any R-labeling. The result uses a structure theorem for R-labelings of the butterfly poset.